Frequency Response of a Differentiator

A differentiator is a circuit that produces an output voltage proportional to the rate of change of the input voltage with respect to time. The frequency response of a differentiator is the plot of the output voltage against the frequency of the input signal.

Linear with Positive Slope

The slope of the frequency response of a differentiator is linear with a positive slope. This means that as the frequency of the input signal increases, the output voltage of the differentiator also increases. The slope of the frequency response is directly proportional to the gain of the differentiator.

Explanation

The frequency response of a differentiator can be derived from the transfer function of the circuit. The transfer function is the ratio of the output voltage to the input voltage in the frequency domain. For a differentiator circuit, the transfer function is given by:

H(jω) = jωRC

where j is the imaginary unit, ω is the angular frequency, R is the resistance, and C is the capacitance of the circuit.

The magnitude of the transfer function is given by:

|H(jω)| = ωRC

This shows that the magnitude of the transfer function is directly proportional to the frequency of the input signal. Therefore, the slope of the frequency response is linear with a positive slope.

Conclusion

In conclusion, the slope of the frequency response of a differentiator is linear with a positive slope. This means that as the frequency of the input signal increases, the output voltage of the differentiator also increases. The slope of the frequency response is directly proportional to the gain of the differentiator.